On the Fundamental Differences between the NT and the KM Center of Sets Calculation Methods on the IT2-FLC Performance

In this paper, we will present the fundamental differences of Nie-Tan (NT) and the Karnik-Mendel (KM) Center of Sets Calculation Methods (CSCMs) on the Interval Type-2 (IT2) Fuzzy Logic Controller (FLC) performance based on analytical derivations. We will derive the Fuzzy Mappings (FMs) of the IT2-FLCs and then investigate how the IT2-FMs are affected by the CSCMs in terms of the Footprint of Uncertainty (FOU) parameters. We will also present a special case where the resulting FM of the KM reduces to its NT counterpart and show that the NT CSCM can be seen as an approximation of the KM CSCM. We will examine three different FOU parameter settings to show that for certain FOU parameter settings the IT2-FLC where the NT CSCMs is employed loses its FOU from a mathematical point of view. We will also present two necessary conditions for the design of the IT2-FLC so that the resulting controller has a symmetrical control surface and is capable to eliminate the steady state error of the system response. Then, by taking account these conditions, we will investigate the gain variations of the IT2-FLCs in terms of the control performance objectives. Based on the observations, we will recommend design guidelines for the IT2-FLCs. The presented results will show that although the NT CSCM has a relatively easier design phase and a closed form representation which might be an enabler for theoretical analyses of the IT2 FLCs, the KM CSCM seems to be superior in overall since it is capable to generate smooth and aggressive control actions which cannot be accomplished by its NT counterpart.